Thirty-Six

Unit 4 · Expressions, Quadratics & PolynomialsA5 · Quadratics · Structures

Polynomial Zeros & Structure

Connect zeros, factors, remainders, and polynomial graphs in hard ACT problems.

  • 30 min
  • Unlocks: Parabolic Bulkheads
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

factor theorem
P(r)=0P(r)=0 exactly when x−rx-r is a factor

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

Translate between roots and factors before expanding anything. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Translate between roots and factors before expanding anything. Keep exact values through the setup; round only when the stem asks.
Core relationship
P(a)=the remainder when P(x) is divided by x−aP(a)=\text{the remainder when }P(x)\text{ is divided by }x-a

In particular, P(a)=0P(a)=0 means x−ax-a divides the polynomial evenly and is therefore a factor.

Algebra or Desmos?

Use grouping, the factor theorem, or synthetic division to expose exact zeros. A graph helps locate possible real zeros, but only algebra confirms a factor and finds any zeros outside the viewing window.

Explore in Desmos

Connect intercepts to polynomial factors

Click all three x-intercepts and connect each zero r to factor x-r.

The calculator is live: change anything and see what happens.

Connect intercepts to polynomial factors

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Trap

The main trap

A root of −3-3 produces x+3x+3, not x−3x-3.

Step 4 of 7Worked examplesDone

Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Factor P(x)=x3−4x2−x+4P(x)=x^3-4x^2-x+4 and find all its zeros.
Group: x2(x−4)−1(x−4)x^2(x-4)-1(x-4).

Which common binomial appears in both grouped terms?

The zeros are −1-1, 11, and 44.

Why does x2−1x^2-1 produce two linear factors?

Worked example

Example 2: ACT twist

Problem

Factor Q(x)=2x3+x2−8x−4Q(x)=2x^3+x^2-8x-4 and find all its zeros.
Group: x2(2x+1)−4(2x+1)x^2(2x+1)-4(2x+1).

Which two zeros come from the factor x2−4x^2-4?

The zeros are −2-2, 22, and −12-\frac12.

Why must both signs be kept when solving x2=4x^2=4?

Faded example · you finish it

Example 3: You finish it

Problem

Find the remainder when x3−2x+5x^3-2x+5 is divided by x−2x-2.
P(2)=8−4+5P(2)=8-4+5

Evaluate 8−4+58-4+5.

Desmos way

Verify the three zeros

Click all three x-intercepts and connect each zero r to factor x-r.

Verify the three zeros

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Step 5 of 7PracticeDone

Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.